can't find ue's on Stich & Korda, sounds like Courier could have had great numbers in these matches from '93 AO(and the final as well)

Courier, 18-0 in sets so far this tournament and a four-set victor over Edberg in last year's final, incredibly made only 10 unforced errors against the 1991 Wimbledon champion Stich. That was just three more than in Courier's most flawless match ever in the previous round against No. 7 Petr Korda.

But it does suggest a lower level of play than in more recent finals, where AM's have been in the 20s-30s, correct?

Also, Brignacca provides the average AM's for the 2005 Grand Slams in the paper you linked in the OP. For men, he has them as follows.

Australian 22.5%
French 10.7%
Wimbledon 29.0%
USO 22.1%

Would it be fair or proper to add or subtract a surface factor when comparing performances in matches played on different surfaces? For example, Rafa's AM in the 2008 RG final was 34.7%, but that might equate to a 53.0% performance at Wimbledon if we added the difference between the French and Wimbledon AM averages.

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Sure, I definitely would think that's fair.

One caveat, though, is that surfaces change over time. The clay at RG in the '90s may well have been slower than it was in '05 when Brignacca got his numbers.

And of course technology changes over time, esp. racquets and strings.

So yeah Rafa's top RG performance should probably be thought of as GOAT-level for clay; so if you imagine the equivalent quality on grass, I guess Nadal's number would be somewhere over 50%.

I wouldn't try to quantify it more precisely than that. Too many variables.

I want to start comparing these AM's that I've posted in distinct pairs -- like comparing two USO finals in consecutive years between the same players. That controls the context and reduces variability in factors like surface, which is definitely a major factor to consider when using AM's.

Nothing too surprising here. Most people who have seen these matches would say, I think, that Safin's level of play was higher than Hewitt's. Not that Hewitt's play was low -- and these figures give a sense of how high it was -- but still, Safin's performance is regularly named as one of the best of the Open Era.

And when you watch the matches you can see plainly that Sampras runs out of gas in '01. Again these figures give some statistical idea of how far his quality dropped.

Nothing too controversial here, just an opening example of how to use AM's to study matches within a controlled context.

Between these two matches, I had always regarded the 1988 final as superior quality. That's because there was somewhat more net play; in '87 the rallies seemed to be longer. It was like watching paint dry, is the way Tony Trabert put it in the booth.

But a few years ago I rewatched the matches, counting winners and service percentages, all the stuff I usually get in my own stats. When I finished watching I actually felt like the '87 final was better played. And the '87 final has higher AM's, as I found out later when I calculated them.

It's true that Lendl was hitting powerful winners all over the place in '88. But he was also making a lot of unforced errors. And his serve really failed him: he made only 42% of first serves -- just 24% in the final set. When he got his serves into the box he made some booming aces; and when he landed his groundstrokes his winners were awesome. That gives the impression of a more exciting match -- which it was. I still would call the '88 final the more exciting one. But I doubt, now, that it had the higher quality.

Lendl was more clutch in '87, especially with his serve. His service percentage never deserted him the way it did in '88; and he actually made some big first serves on a few huge points in the '87 match.

In '87, when everyone was watching the paint dry, there was a stage deep in the long third set when it seemed that neither player could get any kind of advantage. Someone at Tennis Magazine wrote that the two players were fighting over every point, and every game, like two dogs over a bone. And the rallies were often very long, featuring endless backhand exchanges. A lot of people would find that boring, and in some ways it was.

But I remember an exchange in the booth. I think Trabert said that Wilander, despite pulling ahead occasionally, just couldn't consolidate, could not put Lendl away. Newk replied immediately that Lendl couldn't put HIM away either.

And that's the thing: Lendl could not always hit through Wilander. He had the weaponry to do it sometimes, but when Wilander started doing his sprints, Lendl could get into real trouble trying to put Mats away. In other words, he could easily start missing, if he pulled the trigger too soon.

So I think Lendl played well within himself in '87. He couldn't put Mats away so he didn't try to do so prematurely. He was patient enough -- and just barely aggressive enough. He struck the right balance. His patience did make the rallies longer, and could put some people to sleep. But in the end he played his opponent the way he needed to do.

I think the AM's reflect all that. On the other hand, I don't think the AM's reflect too well some of the improvements in Wilander's game in '88. Wilander came to net a lot more in '88; and his service percentage was much higher. His slice, as many people have noted, was working to perfection in '88; it kept Lendl from teeing off, and put him slightly off balance. That had something to do with Lendl making unforced errors; it wasn't JUST a drop in Lendl's quality that was responsible for all his errors.

The AM's do show an improvement by Wilander in '88, though maybe not as much as you would expect to see.

Matchups like Lendl-Wilander imo show, how difficult it is, to determine level of play and to define unforced or forced errors. Both played more on long term strategy, a sort of position war, Lendl a bit too defensive for my gusto, Wilander 1988 more willing to attack on certain points. A match is not decided by the sum of games or points, but by winning the decisive quality points, there are big points and minor significant points. And still the point on the same scoring situation could be more important in different situations. For example: A 40-15 point for a server has different worth than in a 15-40 situation, if the returner is leading.
To the unforced error question. In such a match Lendl- Wilander you could bring up the hyothesis - i now make a too radical argument to get the point over - that they played no unforced errors at all. The same shank is a different error, when hit after one excange or hit after an excange of 30 strokes and more, when the player is near to collapse. So in a way, in such a long match with many extremely long exhausting rallies, many more " easy" appearing errors are "forced" (by nature, exhaustion, mental fatigue and other factors) than in other matches, where a shank is a shank, by lack of concentration, bad positioning, bad stroke production or other pure technical or form factors.

First of all I would like to say that 'aggresive margin" method of calculating players performance is very interesting, although the results are surface - dependent.

Let's assume that all matches are played on the same surface. Does "aggresive margin" method allow us to judge player's performance objectively? Well, it comes close to that, however it has got some drawbacks.

- it assumes that all point are equally important, which is not always the case in tennis; sometimes the player who wins less points than his/her opponent, wins the match,

- the value of 'aggresive margin" for a player in one particular match, depends not only upon the quality of performance [of this player], but also upon the quality of opponents' play,

- is there an objective method to tell unforced errors from forced errors?,

Matchups like Lendl-Wilander imo show, how difficult it is, to determine level of play and to define unforced or forced errors. Both played more on long term strategy, a sort of position war, Lendl a bit too defensive for my gusto, Wilander 1988 more willing to attack on certain points. A match is not decided by the sum of games or points, but by winning the decisive quality points, there are big points and minor significant points. And still the point on the same scoring situation could be more important in different situations. For example: A 40-15 point for a server has different worth than in a 15-40 situation, if the returner is leading.
To the unforced error question. In such a match Lendl- Wilander you could bring up the hyothesis - i now make a too radical argument to get the point over - that they played no unforced errors at all. The same shank is a different error, when hit after one excange or hit after an excange of 30 strokes and more, when the player is near to collapse. So in a way, in such a long match with many extremely long exhausting rallies, many more " easy" appearing errors are "forced" (by nature, exhaustion, mental fatigue and other factors) than in other matches, where a shank is a shank, by lack of concentration, bad positioning, bad stroke production or other pure technical or form factors.

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Without question there are more important points than others, and I'm glad that Moose and I have started consistently counting the number of first serves made on break points, to take one example. In tennis you win if you break in a service game, or if you get the mini-break in a tiebreak. You don't need to win the most points in a match if you're going to win it.

That said, the player who wins the most points, the vast majority of the time, is the player who wins the match. What I like about the AM method is that it always tells you who won the most points; which is the reason that the player with the higher AM almost always wins the match. There is a stronger correlation with the AM than with other common methods which often give the wrong impression about who won the match or about what the margin of victory was (methods like winner/ue differentials and ratios).

When you do come across a match in which the loser has the higher AM, you know automatically that he must have played the important points more poorly. You won't know, of course, which points those were; totals can never point to specific moments. But you will know that the loser's quality of play on important points was lower than the victor's: and that's something you would want to be reflected in any stat, even if the details are something you can only look up by taking a close look at the match.

I have some matches like this. Safin trailed Federer 194-201 in total points at the 2005 AO despite winning the match; so his AM trails Federer's, 19.0% vs 20.8%. The year before there was a similar situation at the 2004 AO, between Safin and Agassi: Safin won fewer points in total, and therefore had the lower AM, but won the match.

Another example is Nadal d. Federer in Rome (the important points lost by Federer in that match can be very simply identified, in that case: he made two UE's on the two match points that he held). Two more examples are Federer d. Agassi at the 2004 USO, and Agassi d. Medvedev in the 1999 RG final, etc.

All of this is to say that the AM, because it tracks the total points won, has a built-in way of telling you who produced the higher quality of play consistently from point to point, over the course of the match (something we would want to know, and which is a strength in any player); and who played better on the important points (something we would also want to know, if it's not the same person who won the most points).

I do think that tennis stats are growing increasingly sophisticated and can focus in on incredibly detailed moments: like telling you how far back Roddick is standing when he returns serve on break points.

Those details are unquestionably important. Tennis writers have always focused on them; it's more difficult to measure them broadly with objective stats, but I think tennis stats are getting there.

As for the argument about not all unforced errors being the same, I can only agree wholeheartedly.

Two strokes may both be wide forehands, and may be executed the same way from a superficial standpoint; but one occurs on the second shot of the exchange, under little pressure, while the other occurs at the end of a wild 30-stroke rally. Unquestionably there's a huge difference.

All that has an impact on the AM's, though I think you can identify the impact and account for it.

Let's take Wilander and Lendl as an example. There you have two great defenders engaged in long rallies. Their AM's in the 1987-88 USO finals were relatively low, because two defenders facing each other will find it hard to hit through each other and pull off winners.

Next year, Wilander loses to Sampras in five sets, and Mats has an AM of 29.8%. Now, was his quality of play really higher than the previous year when he beat Lendl with an AM of 13.8%? Not a chance.

The reason his AM is so high against Sampras is that Sampras came into net relentlessly, forcing to Wilander to make errors or to hit winners: so Mats just didn't have that many chances to make unforced errors. That's the general rule about errors: if you make an error while you're opponent is at the net, your error is typically judged as forced.

The unforced errors that Wilander did make in the Sampras match, when he had a chance to make them, did not come at the end of grueling 50-shot rallies; they were more genuinely "unforced errors" in the ideal sense of that term: unnecessary, sloppy errors.

Knowing all this, you know not to jump to the conclusion that Wilander's level of play in the '89 match was superior to what he did in '88.

I think AM comparisons work best when you can judge AM's that took place on similar surfaces, between opponents of similar style. If there is some radical difference in how the unforced errors occurred, in two matches, then the AM's cannot be used in a straightforward manner, if at all.

Now here is a group of matches without any of the problems discussed above. The venue and surface did not change; the matches took place just one month apart; and the style of play was basically the same, from all three players.

There was a large dropoff in the quality of Federer's play, between the Wimbledon final and the Olympic final. Murray's quality of play was very similar in both matches; and in the Olympic final Murray's level was arguably just as high as what Federer had achieved in the Wimbledon final.

Djokovic experienced a bit of a dropoff at the Olympics -- but not a large one -- if you compare his losses to Federer and to Murray.

Murray, unlike Federer and Djokovic, raised his level of play, going from Wimbledon to the Olympics. And Murray's level of play when he defeated Djokovic was extremely high, almost the highest in the whole group of matches.

The highest performance of all was nevertheless Federer's, in his Wimbledon semifinal over Djokovic.

The roof was closed throughout that match, which is the one change in conditions that I can think of, among these matches. (It was also closed for the last part of the Wimbledon final.) Indoor conditions tend to help Federer; and though I cannot prove it statistically, I would expect AM's to be higher indoors. AM's are generally higher in faster conditions because it's easier to hit winners and to force your opponent into errors.

I would say that AM is a good measure of players' performance because it is relatively simple and it tells you a lot about what happened on a tennis court during a particular match, however this is not an absolute (objective) measure of the match quality (well, perhaps it is, but not a perfect one).

Let's assume that two players played a tennis match. After the match, they calculated their AM's (30% and 20% respectively).

On the second day the players played the second match. The first player performed better than on the first day, so did the second competitor. After the match they calculated their AM's which happened to be exactly the same as in the first match.

Finally, on the third day both players performed better than ever, however their AM's were still the same as previously.

Overall AM's are not always a sensitive measure of player's performance.

On the other hand one could argue that this examples are purely theoretical, without any significant relevance to real tennis matches.

So here's a set of matches all on different surfaces, but it's still possible to use the AM's.

Normally you should see higher AM's on faster surfaces, so in this case you can say that the RG semifinal clearly must have been a better match than the USO final.

I know some posters here have said so; unfortunately I haven't seen the RG match so I can only go by the stats.

Other stats bear this out. The RG and USO meetings both went to five sets. In Paris, Del Potro served 16 aces, a high number for any claycourt match. In New York he served only 8.

In Paris, Federer made 65% of his first serves, a typical number for him. In New York he served at only 50%, his lowest figure to date in all his GS finals; and he made 11 double-faults, a rare instance of his DF's going into double digits.

I would say that AM is a good measure of players' performance because it is relatively simple and it tells you a lot about what happened on a tennis court during a particular match, however this is not an absolute (objective) measure of the match quality (well, perhaps it is, but not a perfect one).

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I agree, it's not an absolute measure; though I doubt that an absolute measure could exist. As Urban has often pointed out, tennis performance is always relative to the opponent. Even if you made all other factors uniform (same surface, same venue, etc.), you will always have two players measuring themselves against each other -- rather than solely against an absolute measure.

Don't get me wrong, I do think there are some absolute measures. How fast you can hit a serve is one. I just don't think it's possible to measure exclusively against absolute standards, without reference to the opponent (not that this is your argument).

Let's assume that two players played a tennis match. After the match, they calculated their AM's (30% and 20% respectively).

On the second day the players played the second match. The first player performed better than on the first day, so did the second competitor. After the match they calculated their AM's which happened to be exactly the same as in the first match.

Finally, on the third day both players performed better than ever, however their AM's were still the same as previously.

Overall AM's are not always a sensitive measure of player's performance.

On the other hand one could argue that this examples are purely theoretical, without any significant relevance to real tennis matches.

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I've wondered about this too -- whether two players can increase their level of play, from one match to the next, without increasing their AM's. But I'm skeptical that it can happen.

If two players meet on one day and have a rematch the next day, and both players improve their performance, they will probably say something like this to each other:

"You know, yesterday we played like crap, we were spraying the ball all over the place. Today we hit with a lot more control; I felt like I hardly made any unforced errors. You forced me into some errors, because you played better, too: but you had to force me into them. I wasn't giving you any free points, and you weren't giving me any."

If all that is true -- if the UE's go down -- the AM's have to go up.

It's an interesting question, though. Can a scenario be described in which two players are hitting the ball better, in the rematch, but their AM's stay the same?

I thought that Del Potro's effort in the last 2 sets of that match was pretty pathetic. Guess the AM showed that.

What was Connors' AM in the '84 W Final?

But a few years ago I rewatched the matches, counting winners and service percentages, all the stuff I usually get in my own stats. When I finished watching I actually felt like the '87 final was better played.

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Taking stats can change your impression of a match. Thought much more of the '87 FO final after taking stats on it.

IAs Urban has often pointed out, tennis performance is always relative to the opponent. Even if you made all other factors uniform (same surface, same venue, etc.), you will always have two players measuring themselves against each other -- rather than solely against an absolute measure.

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Here's a good example. I used stats from Moose and yourself in old threads for two consecutive French Open finals:

So here's a set of matches all on different surfaces, but it's still possible to use the AM's.

Normally you should see higher AM's on faster surfaces, so in this case you can say that the RG semifinal clearly must have been a better match than the USO final.

I know some posters here have said so; unfortunately I haven't seen the RG match so I can only go by the stats.

Other stats bear this out. The RG and USO meetings both went to five sets. In Paris, Del Potro served 16 aces, a high number for any claycourt match. In New York he served only 8.

In Paris, Federer made 65% of his first serves, a typical number for him. In New York he served at only 50%, his lowest figure to date in all his GS finals; and he made 11 double-faults, a rare instance of his DF's going into double digits.

Here's a good example. I used stats from Moose and yourself in old threads for two consecutive French Open finals:

1978
Borg 10.53%
Vilas -15.8% (negative AM)

1979
Borg 31.35%
Pecci 16.3%

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That's a good example, though I have one major caveat, and a few minor ones.

When Borg played Vilas, he deliberately did nothing more than rally endlessly with him. He knew Vilas would eventually make the error, so why go for more? And Borg had no concerns about stamina, so that's what he did.

In other words, though I think you're right that Borg's AM would have been relatively low no matter what, due to the styles involved -- I think it still has to be said that Borg's best tennis was not on display. Pecci unquestionably forced him to do more and to dig deeper.

Minor comments: the UE's in '78 are my own count; the UE's in '79 come from Bud Collins. I doubt that it makes much difference, but whenever the statisticians change from match to match I like to note it.

I don't know whether Bud included Pecci's 3 double-faults (Borg made none). My guess is that he did. So going with his numbers, Borg made 17 UE's, Pecci 59. Borg led in total points by 145-107, so this is what I get for the AM's:

Now here is a group of matches without any of the problems discussed above. The venue and surface did not change; the matches took place just one month apart; and the style of play was basically the same, from all three players.

There was a large dropoff in the quality of Federer's play, between the Wimbledon final and the Olympic final. Murray's quality of play was very similar in both matches; and in the Olympic final Murray's level was arguably just as high as what Federer had achieved in the Wimbledon final.

Djokovic experienced a bit of a dropoff at the Olympics -- but not a large one -- if you compare his losses to Federer and to Murray.

Murray, unlike Federer and Djokovic, raised his level of play, going from Wimbledon to the Olympics. And Murray's level of play when he defeated Djokovic was extremely high, almost the highest in the whole group of matches.

The highest performance of all was nevertheless Federer's, in his Wimbledon semifinal over Djokovic.

The roof was closed throughout that match, which is the one change in conditions that I can think of, among these matches. (It was also closed for the last part of the Wimbledon final.) Indoor conditions tend to help Federer; and though I cannot prove it statistically, I would expect AM's to be higher indoors. AM's are generally higher in faster conditions because it's easier to hit winners and to force your opponent into errors.

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After thinking about it for a few days, I think their is some issue with this reasoning. The AM is the measure of the ability of a player to execute successfully an aggressive pattern of winning points. From our spectator's point of view, we will often equate an high AM with high quality of play, because winners are more spectacular . Thus, we will consider a match where one player as high AM as a pleasant match, and more so if both players have an high AM. Thus, quality means more, is this case, spectacular than "high competitive level of play".

For exemple, I disagree with your statement that the level of play of Murray in the Olympic final is as high as Fed's level during the Wimbledon final. We know that he was able to execute an aggressive pattern of winnings points as successfully as Roger did, and we know than from a spectator standpoints, his performance was as enjoyable. But that doesn't necessarily means that he played better, as the AM is related to the level of his opponent. In this case, the AM of Federer for the Olympic final suggest that he played worse: he couldn't execute successfully an aggressive pattern of play, and he probably didn't defend as well, thus enabling Murray to considerably rise his own AM, even if himself played worse too! The inverse is possible too: Murray was so good in the Olympic final that Federer's AM dropped. I know for seeing these matches that Federer's level did drop off (he was clearly mentally tired after is SF against Del Potro), but the statistics themselves don't really tell the competitive level of play of both players. They say that the Wimbledon final was a better match from a spectator points of view.

I think federer served exceptionally brilliantly in the semis, but he did return better in the finals ..

murray in the finals also returned quite a bit better than djoker did in the semis ...

the matches were closer in quality than the respective AMs seem to suggest ...

differences being the roof and the better quality of returning resulting in more rallies and as a result more UEs in the final

I will admit, I am a bit surprised that they have federer at only 10 UEs in the semi. I thought he made more than that ...

at olympics :

I think djoker and murray played more or less at the same levels as they did @ wimbledon .... federer of course played much worse in the finals ( his AM there reflects that ) ... of course as a result of federer's UEs, murray's AM goes down in the finals as compared to the semis ... I do think he played slightly better in the finals ....

If we bring the high AMs in the Olympics semis b/w fed and delpo also in here, well IMO that really flatters the quality of that match ... both were serving well and being aggressive , but returning poorly and not moving that well ( even federer, by his standards )

I can't get reliable UE counts for any of them. In the 1993 final I'm missing UE counts for both Sampras and Courier. In all of the rest of Pete's finals I have his UE count but no UE count for his opponents, without which I can't calculate AM's.

I think federer served exceptionally brilliantly in the semis, but he did return better in the finals ..

murray in the finals also returned quite a bit better than djoker did in the semis ...

the matches were closer in quality than the respective AMs seem to suggest ...

differences being the roof and the better quality of returning resulting in more rallies and as a result more UEs in the final

I will admit, I am a bit surprised that they have federer at only 10 UEs in the semi. I thought he made more than that ...

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I agree with a lot of this, though I would question whether the lower AM's in the final were due to better returning.

This gets to the issue we've been debating in recent days in this thread, about whether it's possible for two players to lift their play, from one match to the next, but for their AM's to stay the same or even to go lower.

I doubt that better returning can result in lower AM's. Let's say I make 20 unforced errors on the return, in one match. I cut that down to 10 in the rematch. The quality of play has gone up, and my AM will be higher, because I've cut down on my UEs.

You said that the better returning produced a greater number of rallies, and that long rallies tend to end in UEs. I agree that long rallies often end in UEs. But not all of them. Some of those points -- in fact, a decent number, even among insanely good defenders like Nadal and Djokovic -- will end in winners, or in forcing shots.

Let's do the math. On the return, I reduced my UE's by 10. Yesterday on those 10 points I didn't put the ball back in play; today I made rallies out of those 10 points. Let's say I make UE's on 6 of those rallies. I still have an overall improvement of 4 UE's.

If I reduced my return UE's by 10, and ALL TEN rallies ended in UE's, then my AM would stay the same. But the chances of that happening are small. If even one of those points ends with me making a winner or a forcing shot, then I've come out ahead overall, and my AM will rise from the previous day.

As for the real matches at Wimbledon, I'm not sure that the returning was better in the final than in the semi.

Unfortunately I only saw the middle of Fed-Djokovic. (I saw all of the final). I did notice Djokovic having trouble with Federer's serve, and I wondered whether Djokovic was just having an off day, or whether his trouble was due to Federer's serving. I really thought Djokovic looked handcuffed by Federer's serve, for the most part.

Wimbledon.org had Djok making only 8 UE on the return, which is pretty good returning in a four-set match. They had Federer making only 5 UE on the return -- so Federer really could not have returned much better, if at all, in the final.

Fed/Djok had the roof closed throughout. Federer's serve traveled faster by a few mph whenever they closed the roof. In the final, when they closed the roof one of the announcers said that his average 1st serve went up by 5 mph.

But the Fed/Djok semi was played entirely under the roof, and if you look at the Wimbledon boxscores Federer did have more mph on his serves in the semifinal, as compared to the final. His average 1st serve was 2 mph higher in the semis. His average 2nd serve was 4 mph higher, which is a significant margin -- and that partly explains how Federer managed to win 72% of his 2nd serve points in that match. That's just phenomenal; I fully agree with you that he served extremely well against Djok.

But I think that means that Djokovic looked like he was returning worse than Murray, because Djokovic had to deal with better serves.

The same seems to be true on the other side of the coin. Did Federer improve his returns, going from the semis to the finals? Or did his returning just look worse against Djokovic, because Novak was serving better than Andy?

I think Novak was serving better. Perhaps the indoor conditions helped the mph on his serve. Murray barely had a faster 1st serve, on average, than Djokovic (121 mph vs 120 mph). But on second serve Djokovic was clearly superior: 95 mph vs 88 mph.

So Federer's returning may have looked better in the final because he had more of a puffball to deal with, on second serves.

7 mph is a significant difference. And yet Federer, dealing with these faster second serves from Djokovic, made only 5 ue's errors on the return.

Federer cannot have cut that number down by much, if at all, in the final.

So I think if the AM's are lower in the final than in the semifinal, we may need to look for another cause.

The mph figures, and the return UE's counts, actually suggest that the semifinal had better serving and returning than the final did.

After thinking about it for a few days, I think their is some issue with this reasoning. The AM is the measure of the ability of a player to execute successfully an aggressive pattern of winning points. From our spectator's point of view, we will often equate an high AM with high quality of play, because winners are more spectacular . Thus, we will consider a match where one player as high AM as a pleasant match, and more so if both players have an high AM. Thus, quality means more, is this case, spectacular than "high competitive level of play".

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Actually the AM's cannot tell you which matches were most entertaining -- if we think of entertaining matches as having a lot of winners.

There are two ways to get a high AM. You could increase your winners; that would be a more entertaining match. Or you could get a high AM by decreasing your unforced errors. In the latter case, maybe the number of winners you made was not unusually high and what was really great about your level of play was how few errors you made. That sort of match would not qualify as entertaining -- not if we're looking for winner-heavy matches.

Maybe you read my post above to Urban, about the 1987-88 USO finals between Wilander and Lendl. Most people, myself included, would consider the '88 final to be more exciting. There is more net play and Lendl is hitting a lot of passes and other aggressive winners. The rate of winners -- meaning how frequently they occurred in the match -- was faster than in the '87 match. But the rate of errors was also faster than it was in '87.

The '87 final featured endless rallies, and some people said it was like watching paint dry. I actually think the match was better quality than the '88 final, because Lendl's UE's were so low. (He went from 58ue in '87 to 85ueE in '88.) But that made the match less exciting.

And you can guess what I'm going to say next: the less exciting match had the higher AM's.

For exemple, I disagree with your statement that the level of play of Murray in the Olympic final is as high as Fed's level during the Wimbledon final. We know that he was able to execute an aggressive pattern of winnings points as successfully as Roger did, and we know than from a spectator standpoints, his performance was as enjoyable. But that doesn't necessarily means that he played better, as the AM is related to the level of his opponent. In this case, the AM of Federer for the Olympic final suggest that he played worse: he couldn't execute successfully an aggressive pattern of play, and he probably didn't defend as well, thus enabling Murray to considerably rise his own AM, even if himself played worse too! The inverse is possible too: Murray was so good in the Olympic final that Federer's AM dropped. I know for seeing these matches that Federer's level did drop off (he was clearly mentally tired after is SF against Del Potro), but the statistics themselves don't really tell the competitive level of play of both players. They say that the Wimbledon final was a better match from a spectator points of view.

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The part I bolded is where I have disagreements.

Federer did not play well in the Olympic final, agreed. He made a lot more UE's than in the Wimbledon final, yes, so far so good.

But if your opponent's UE's rise, your own AM will fall. You can work it out in Excel or with a calculator: punch in higher numbers for Federer's UE. Murray's AM will fall.

Or just think about it this way: when your opponent's UEs increase, you're getting more of the points for free. More of the points in the match are being handed to you on a platter (I'm exaggerating to make the point) -- that is, you did not have to make any aggressive plays on those points. The number of points that you have to win through your own aggressive plays to win the match has suddenly decreased.

Let's take the Olympic final and increase Federer's UEs.

Federer won 68 points and made 31 UE's - AM of 12.1%
Murray won 97 points and made 17 UE's - AM of 29.7%

Those were the real numbers. Murray won 97 points in the match, making aggressive plays for 66 of the points. And he won 31 additional points that Federer handed to him.

Now let's change Federer's UE to 41:

Federer won 68 points and made 41 UE's - AM of 6.1%.
Murray won 97 points and made 17 UE's - AM of 23.6%.

Now Murray only has to make aggressive plays on 56 points; 41 points are now being handed to him.

Let's say Murray tanks, and hands Federer all of the points in the match. Federer wins 6-0, 6-0, 6-0. Eighteen straight games, 72 straight points -- all of them won by Federer on Murray's unforced errors. Federer didn't have to lift a finger to win anything.

Murray's AM in that scenario would be negative: -100%. Federer's AM would be exactly 0.0%.

I think you've put your finger on some good issues, but I don't think it's valid to say that Murray's AM was high in the Olympic final because Federer's was low. If your opponent is handing you the match, you're getting free points rather than making aggressive plays, so your own AM has to go down too.

I agree with a lot of this, though I would question whether the lower AM's in the final were due to better returning.

This gets to the issue we've been debating in recent days in this thread, about whether it's possible for two players to lift their play, from one match to the next, but for their AM's to stay the same or even to go lower.

I doubt that better returning can result in lower AM's. Let's say I make 20 unforced errors on the return, in one match. I cut that down to 10 in the rematch. The quality of play has gone up, and my AM will be higher, because I've cut down on my UEs.

You said that the better returning produced a greater number of rallies, and that long rallies tend to end in UEs. I agree that long rallies often end in UEs. But not all of them. Some of those points -- in fact, a decent number, even among insanely good defenders like Nadal and Djokovic -- will end in winners, or in forcing shots.

Let's do the math. On the return, I reduced my UE's by 10. Yesterday on those 10 points I didn't put the ball back in play; today I made rallies out of those 10 points. Let's say I make UE's on 6 of those rallies. I still have an overall improvement of 4 UE's.

If I reduced my return UE's by 10, and ALL TEN rallies ended in UE's, then my AM would stay the same. But the chances of that happening are small. If even one of those points ends with me making a winner or a forcing shot, then I've come out ahead overall, and my AM will rise from the previous day.

As for the real matches at Wimbledon, I'm not sure that the returning was better in the final than in the semi.

Unfortunately I only saw the middle of Fed-Djokovic. (I saw all of the final). I did notice Djokovic having trouble with Federer's serve, and I wondered whether Djokovic was just having an off day, or whether his trouble was due to Federer's serving. I really thought Djokovic looked handcuffed by Federer's serve, for the most part.

Wimbledon.org had Djok making only 8 UE on the return, which is pretty good returning in a four-set match. They had Federer making only 5 UE on the return -- so Federer really could not have returned much better, if at all, in the final.

Fed/Djok had the roof closed throughout. Federer's serve traveled faster by a few mph whenever they closed the roof. In the final, when they closed the roof one of the announcers said that his average 1st serve went up by 5 mph.

But the Fed/Djok semi was played entirely under the roof, and if you look at the Wimbledon boxscores Federer did have more mph on his serves in the semifinal, as compared to the final. His average 1st serve was 2 mph higher in the semis. His average 2nd serve was 4 mph higher, which is a significant margin -- and that partly explains how Federer managed to win 72% of his 2nd serve points in that match. That's just phenomenal; I fully agree with you that he served extremely well against Djok.

But I think that means that Djokovic looked like he was returning worse than Murray, because Djokovic had to deal with better serves.

The same seems to be true on the other side of the coin. Did Federer improve his returns, going from the semis to the finals? Or did his returning just look worse against Djokovic, because Novak was serving better than Andy?

I think Novak was serving better. Perhaps the indoor conditions helped the mph on his serve. Murray barely had a faster 1st serve, on average, than Djokovic (121 mph vs 120 mph). But on second serve Djokovic was clearly superior: 95 mph vs 88 mph.

So Federer's returning may have looked better in the final because he had more of a puffball to deal with, on second serves.

7 mph is a significant difference. And yet Federer, dealing with these faster second serves from Djokovic, made only 5 ue's errors on the return.

Federer cannot have cut that number down by much, if at all, in the final.

So I think if the AM's are lower in the final than in the semifinal, we may need to look for another cause.

The mph figures, and the return UE's counts, actually suggest that the semifinal had better serving and returning than the final did.

And if so, that's reflected in the AM's.

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firstly, I think federer *did* make more unforced errors than what is shown in the semi stats ... maybe I'd have to watch the match again ..

again, just the UEs made on the return aren't the only measure of the quality of the return, right ? its ok, if you make a few unforced errors here and there if you are trying to attack , but if you can put pressure on the server by attacking, that'd have achieved the purpose.

Also, if a player is returning well, points that could've been forced errors on the return get back into play and those rallies could end in unforced errors

only in the first set, speed was less, so I don't think it had to do with the roof .... probably had to do with some back pain/ache for federer or his back being stiff - he removed the extra shirt mid-way through the second set - probably because he felt better ..

federer had back problems in the benneateau match and only upped the ante on his serve when he had to, after being down two sets to love .... then in the malisse match, it was obvious that even his movement was hindered because of the back problem ...given that I was surprised at how well he served in the djokovic match

federer did obviously find it easier vs murray's 2nd serve, but first serve ? I think murray's was better ... not that much of a difference in federer's returning in the semis and finals, but I think he was a bit better in the finals than in the semis

anyways djoker's returning did look worse because federer served quite a bit better in the semis, but I do think the other part of it was that murray did in fact return quite a bit better ( even if it was against a worse serve )

I remember murray going deep into almost every service game of federer's, but in the djoker match, I remember several games that just flew by, at love or 15 ...

Perhaps AM should be viewed as a way to describe a tennis match rather than as a measure of its quality?

The ultmate goal in tennis is to win, isn't it?

So Borg beat Vilas in RG 1978 final because of his consistency and his stamina (perseverence, patience, discipline?) - both are very important for a tennis player.

He used his weapons to beat his opponent, however he employed a bit unusual tactics.

Should his performance be regarded as inferior (when compared to some of his other matches) because he decided to use less spectacular method to win?

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No it shouldn't necessarily be regarded as inferior just because Borg employed a less spectacular method. You can have a high AM by lowering your UE's and playing un-spectacular tennis.

Lendl, for example, had a lower UE count in his '87 USO final than he did in the '88 final (both against Wilander). He also struck fewer winners in '87, because he was employing, as you put it, a less spectacular strategy.

But Lendl's AM is higher in '87. So the AM is not punishing the less spectacular performance. In this case the less spectacular is actually being rewarded.

The Borg/Vilas/Pecci example is trickier, because Borg’s less spectacular performance against Vilas does have a lower AM than his performance against Pecci.

That's because Borg actually made more UE's against Vilas. In that very short match (short in terms of number of points played), Borg made 19 UE, by my count. In the much longer Pecci match, Borg made only 17 UE (Bud Collins' count).

It's difficult to use AM's when the players you're comparing have radically different styles. Pecci was always coming into net, always forcing the issue. Borg had fewer opportunities to make an UE, when facing Pecci. On many points Borg was either forced into an error, or he made a winning pass. When Borg played Vilas, almost every point was an opportunity to make an UE because both players stayed back and rallied endlessly.

But it was a smart way to play Vilas -- at least for Borg.

So was Borg's '78 performance inferior to '79? I'm not certain. Borg was in peak form both years, and '78 was actually the year that he set the record for fewest games lost in the tournament.

On the one hand, I know that Borg had more UE's errors in '78 not because he played below form but because long rallies against another baseliner present constant opportunities to make UE's.

On the other hand I don’t think that Vilas forced Borg to dig deep or to do anything special. Against Pecci, Borg had to run more, make precision passes, keep Pecci from attacking his serve, etc.

I might have to say that Borg’s performance against Pecci was the higher performance. It’s not specifically his higher AM against Pecci that makes me say that, because AM's are a bit deceptive in comparisons that involve radically different styles. I say that just because I think that Pecci played a better match than Vilas. Pecci was in better form in ’79 than Vilas was in ’78. And in fact Pecci beat Vilas on the way to the '79 final, in straight sets. Pecci was a very good claycourter in peak form, and he could get "hot." Vilas in ’78 was struggling a bit with his game – and had a bit of a mental block against Borg.

Stich did not play that badly (he actually had fewer UE's than Sampras); but he got straight-setted because Sampras played so well.

Yet it's still surprising, because Sampras said that Stich was one of his most difficult opponents.

I think when you give your UE count, your terms "FH" and "BH" include both ground strokes and net strokes?

I presume that both players came in behind all of their serves, so there cannot have been many opportunities for either player to make unforced errors with groundstrokes.

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I'm curious as to whether you think the issue implicit in the bolded section of your post undermines the usefulness of AM as a means of measuring quality of play. It seems to me that in matches where both players regularly attack the net, their respective AMs will tend to be higher than those of players who stick more to the baseline. The reason for this, as you alluded to, is that when a player approaches the net he takes away any opportunity for his opponent to hit an unforced error, because attempted passing shot errors are counted as forced by convention. This effectively halves the probability that an unforced error is hit during a point where one player is at net when compared to a point where both players remain at the baseline, as in the former only one player can potentially make an unforced error, while in the latter either player can. This fact is important because any reduction in unforced errors necessitates a corresponding increase in the fraction of points won by aggressive plays, and hence an increase in AM.

Was the Sampras-Stich match really as high quality as the AMs make it out to be, for instance? I have my doubts; it seems that these figures might, to some extent, be an artifact of the playing styles of Sampras and Stich.

I feel this is an issue, not least of all because I see no reason to think that matches involving lots of serve and volleying or lots of chip and charging are of higher quality in general than matches played primarily from the baseline.

I'm curious as to whether you think the issue implicit in the bolded section of your post undermines the usefulness of AM as a means of measuring quality of play. It seems to me that in matches where both players regularly attack the net, their respective AMs will tend to be higher than those of players who stick more to the baseline. The reason for this, as you alluded to, is that when a player approaches the net he takes away any opportunity for his opponent to hit an unforced error, because attempted passing shot errors are counted as forced by convention. This effectively halves the probability that an unforced error is hit during a point where one player is at net when compared to a point where both players remain at the baseline, as in the former only one player can potentially make an unforced error, while in the latter either player can. This fact is important because any reduction in unforced errors necessitates a corresponding increase in the fraction of points won by aggressive plays, and hence an increase in AM.

Was the Sampras-Stich match really as high quality as the AMs make it out to be, for instance? I have my doubts; it seems that these figures might, to some extent, be an artifact of the playing styles of Sampras and Stich.

I feel this is an issue, not least of all because I see no reason to think that matches involving lots of serve and volleying or lots of chip and charging are of higher quality in general than matches played primarily from the baseline.

Do you have any thoughts on this?

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Yes I think this is an issue with any stats that use Unforced Errors: namely methods like the Aggressive Margin, winner/error differentials, ratios, etc.

On grass you will commonly see very high winner/error differentials: +50 or more. On clay it's hard to get even a positive differential, and negative ones are common.

Perhaps I should have listed the AM's grouped by surface, because that is the factor that seems to have the greatest impact on how people play -- and therefore an impact on how UE's are scored in those matches.

On grass two SVers coming in behind every serve will have very few opportunities to make UE's. On clay two baseliners engaged in a war of attrition are going to have many such opportunities.

That's one reason that AM's are higher on grass. But leaving aside the question of playing styles, it's also just generally true that you can make more winners and forcing plays on a fast surface. That has nothing to do with styles per se, because you could have two baseliners playing each other, and they will find it to be true: they will have an easier time forcing their opponent when they're playing on a fast surface.

On clay you can hit the ball just as hard, or just as wide, but the defender gains an advantage, as compared to grasscourt matches. Any surface that slows down the ball will make it easier for the defender to catch up to it, and harder for anyone to hit winners or to make forcing plays.

And that's what the AM is about: making forcing plays while keeping your unforced errors down. That is just inherently easier on a fast court.

So that makes it really impossible to compare AM's -- or winner/error differentials and ratios -- across surfaces (except for limited purposes like I did with the Federer-Delpo matches).

Take the top AM's in my opening post as examples. McEnroe has 53% on grass, the highest AM of all. The highest AM on clay belongs to Nadal, with 35%. No one would suggest that McEnroe's level on grass is a greater level than what Nadal achieved on clay. I think it's fair to assume that both of those performances were GOAT level for their respective surfaces.

Sampras/Stich should be compared to other grasscourt matches of the time period. It was very common then for players to follow all, or nearly all, of their serves to net. Sampras/Stich could easily be compared with Sampras/Becker, for example.

But if you try comparing it to this era's matches at Wimbledon, which commonly feature two baseliners slugging it out, you're going to have problems.

At the very least you could not use the numbers in such a comparison literally, if at all.

But that all results from surface being such a large factor in tennis. I'm not sure there's any good or efficient way around this. I, for one, would not want to chuck the distinction between a forced and an unforced error; I would just restrict comparisons by surface.

I rewatched this match ...I take back my statement that federer returned better in the final .. I think he was just as good in the semis as he was in the finals as far as returning goes ... However still think murray did return quite a bit better than djoker ...( even though it was against a weaker federer serve )

at the tail end of the match, a stat was shown, 57% returns in for djoker and 68% in for federer

I did the UE stats for this as well,

wimbledon.org has UEs by set :

federer : 2,3,4,1 - total of 10
djokovic : 5,3,9,5 - total of 22

I have significantly different :

federer : 4,4,9,1 - total of 18
djokovic : 6,3,11,7 - total of 27

which brings their AMs to :

federer : 31.95%
djokovic : 26.4%

funnily enough, I have 4 UEs on the return for fed,7 for djoker
wimbledon.org has 5 for fed and 8 for djoker

so I don't know how lenient the statistician was on strokes apart from the return :shock:

My initial impression was justified, there were more unforced errors than what is on the site and difference is non-trivial ..

If by any chance, you are planning to watch this, can you please keep track of the UEs and see how many you end up with ?

I rewatched this match ...I take back my statement that federer returned better in the final .. I think he was just as good in the semis as he was in the finals as far as returning goes ... However still think murray did return quite a bit better than djoker ...( even though it was against a weaker federer serve )

at the tail end of the match, a stat was shown, 57% returns in for djoker and 68% in for federer

I did the UE stats for this as well,

wimbledon.org has UEs by set :

federer : 2,3,4,1 - total of 10
djokovic : 5,3,9,5 - total of 22

I have significantly different :

federer : 4,4,9,1 - total of 18
djokovic : 6,3,11,7 - total of 27

which brings their AMs to :

federer : 31.95%
djokovic : 26.4%

funnily enough, I have 4 UEs on the return for fed,7 for djoker
wimbledon.org has 5 for fed and 8 for djoker

so I don't know how lenient the statistician was on strokes apart from the return :shock:

My initial impression was justified, there were more unforced errors than what is on the site and difference is non-trivial ..

If by any chance, you are planning to watch this, can you please keep track of the UEs and see how many you end up with ?

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I re-watched this match a couple days ago and had the same reaction I had the first time I watched and saw the very low UE count flash on screen: What!?

BTW, I didn't score it because I don't know how. Can anyone suggest an authoritative guide to differentiating UE, winners and forced errors?

Those stats are from Wimbledon.com and the New York Times tennis blog; I also used ABMK's late-match graphic.

Final
Murray got back 99 of Federer’s 128 good serves: 77.3%
Federer got back 107 of Murray’s 156 good serves: 68.6%

I got stats for the final by using a graphic that was shown in the fourth set. At 2-all, 15-love, right after Murray forces a BH return error, the graphic had Murray with 43 unreturned serves and Federer with 25. I counted to the end – including aces in my count, which I think is the way this is done though I'm not absolutely certain – and finished with Murray at 49 and Federer 29.

The graphic makes it look like Murray was doing nearly twice as well as Federer in serving unreturned serves. That's because it's a simple count of how many serves were unreturned; it does not take into account the fact that Murray served a lot more points in the match than Federer did. When you convert the counts to the percentages above, there was not much of a difference between the two players.

ABMK, even though Federer had more mph on his serves in the semifinal, compared to what he served at Murray, I think these numbers support your argument that Murray returned better than Djokovic (btw I'll have more extensive replies to your posts later).

Federer's second serve was really humming in the semifinal, so if anyone counts the return errors in this match it would be useful to know how many of his second serves went unreturned. If a LOT of Djokovic's return errors occurred on second serve then we might have to give more credit to Federer for making Djokovic's returning look worse than Murray's.

We have been debating whether it is possible for the two players to raise their level of play without changing the AM's.

I would like to present a hypothetical scenario in which two players playing on two occasions on the same surface exhibit different level of play yet their calculated AM’s stay the same.

It is open to debate if such a scenario is realistic.

On the first day they played an average match; winners, forced and unforced errors were counted, and the AM's were calculated.

On the second day they played an excellent match, however the number of winners, forced and unforced errors was the same as on the first day (for the sake of simplicity I assume that the number of points in both matches was the same). How is it possible?

Winners;
The first player was hitting great shots, however the second player was also defending better than previously (had the second player been playing as well as on the first day, the first player would have had more winners than on the first day, had the first player been playing as well as in the first match he would have had less winners than in the first match). As a result the first player had the same number of winners on both occasions, and the same applies to the second player.

Forced errors;
Generally the same logic applies to forced errors. Both players committed the same number of forced errors as during the first match, however the shots that forced these errors were of the higher quality than the shots that forced errors in the first match.

Unforced errors;
Generally the same logic applies to unforced errors. Both players committed the same number of unforced errors as during the first match, however the points which ended in unforced errors were of the higher quality than points which ended in unforced errors in the first match. We have already mentioned that points that end in an unforced error may be of different quality (e.g. long neutral rallies versus double faults or return errors or short neutral rallies).

Yes I think this is an issue with any stats that use Unforced Errors: namely methods like the Aggressive Margin, winner/error differentials, ratios, etc.

On grass you will commonly see very high winner/error differentials: +50 or more. On clay it's hard to get even a positive differential, and negative ones are common.

Perhaps I should have listed the AM's grouped by surface, because that is the factor that seems to have the greatest impact on how people play -- and therefore an impact on how UE's are scored in those matches.

On grass two SVers coming in behind every serve will have very few opportunities to make UE's. On clay two baseliners engaged in a war of attrition are going to have many such opportunities.

That's one reason that AM's are higher on grass. But leaving aside the question of playing styles, it's also just generally true that you can make more winners and forcing plays on a fast surface. That has nothing to do with styles per se, because you could have two baseliners playing each other, and they will find it to be true: they will have an easier time forcing their opponent when they're playing on a fast surface.

On clay you can hit the ball just as hard, or just as wide, but the defender gains an advantage, as compared to grasscourt matches. Any surface that slows down the ball will make it easier for the defender to catch up to it, and harder for anyone to hit winners or to make forcing plays.

And that's what the AM is about: making forcing plays while keeping your unforced errors down. That is just inherently easier on a fast court.

So that makes it really impossible to compare AM's -- or winner/error differentials and ratios -- across surfaces (except for limited purposes like I did with the Federer-Delpo matches).

Take the top AM's in my opening post as examples. McEnroe has 53% on grass, the highest AM of all. The highest AM on clay belongs to Nadal, with 35%. No one would suggest that McEnroe's level on grass is a greater level than what Nadal achieved on clay. I think it's fair to assume that both of those performances were GOAT level for their respective surfaces.

Sampras/Stich should be compared to other grasscourt matches of the time period. It was very common then for players to follow all, or nearly all, of their serves to net. Sampras/Stich could easily be compared with Sampras/Becker, for example.

But if you try comparing it to this era's matches at Wimbledon, which commonly feature two baseliners slugging it out, you're going to have problems.

At the very least you could not use the numbers in such a comparison literally, if at all.

But that all results from surface being such a large factor in tennis. I'm not sure there's any good or efficient way around this. I, for one, would not want to chuck the distinction between a forced and an unforced error; I would just restrict comparisons by surface.

But maybe we need to think more about how to score UE's.

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Your point about surface affecting AM is well taken. You say, for instance, that fast/low-bouncing surfaces tend to have higher AMs. The reason you give for this seems to be twofold: (1) it's easier to execute aggressive plays on fast/low-bouncing surfaces; and (2) netplay is more common on fast/low-bouncing surfaces. Both of these two aspects (easier to force plays, more netplay) seem to positively affect AM, yet you talk as if matches played on the same surface (with the same speed) are roughly comparable with respect to AM. This would only be true if the amount of netplay is roughly constant between matches on a given surface.

It's not clear to me that their is this constancy - especially if we look across eras. Compare Borg-Vilas or Djokovic-Nadal at the FO to Laver-Rosewall. The former pairs remained almost exclusively at the baseline, while the latter approached the net in most points. I include Djokovic-Nadal because, while both camped out on the baseline, they also both go for winners when they have an opening, perhaps in contrast to the more conservative play of Borg-Vilas. This is worth mentioning because I'm going to guess Laver-Rosewall would have the highest AMs of the three pairs due to less unforced errors, but I don't want this attributed to the 'rally-all-day' style of Borg-Vilas. Rather, I think, Laver-Roswall will have the higher AMs than the other pairs because they had less opportunity to hit unforced errors due to the frequent net approaches. This would be the case even though the surface is held relatively constant.

You've talked about this yourself, and the lesson I think is that one can only use AM to compare match quality when holding the surface and level of net play relatively constant. Otherwise AMs can be easily inflated or deflated by factors unrelated to quality of play.